Our Knowledge of Numbers as Self-Subsistent Objects

Abstract

†A feature of Frege’s philosophy of arithmetic that has elicited a great deal of attention in the recent secondary literature is his contention that numbers are ‘self-subsistent’ objects. The considerable interest in this thesis among the contemporary philosophy of mathematics community stands in marked contrast to Kreisel’s folk-lore observation that the central problem in the philosophy of mathematics is not the existence of mathematical objects, but the objectivity of mathematics. Although Frege was undoubtedly concerned with both questions, a goal of the present paper is to argue that his success in securing the objectivity of arithmetic depends on a less contentious commitment to numbers as objects than either he or his critics have supposed. As such, this paper is an articulation and defense of both Frege’s analysis of arithmetic and Kreisel’s observation.